def test_update_H(dim, subdim, method, symm, pd):
rng = np.random.RandomState(1)
tol = dict(atol=1e-6, rtol=1e-6)
B = get_matrix(dim, dim, pd, True, rng=rng)
H = get_matrix(dim, dim, pd, True, rng=rng)
S = get_matrix(dim, subdim, rng=rng)
Y = H @ S
B1 = update_H(None, S, Y, method=method, symm=symm)
np.testing.assert_allclose(B1 @ S, Y, **tol)
B2 = update_H(B, S, Y, method=method, symm=symm)
np.testing.assert_allclose(B2 @ S, Y, **tol)
if subdim == 1:
B3 = update_H(B, S.ravel(), Y.ravel(), method=method, symm=symm)
np.testing.assert_allclose(B2, B3, **tol)
B4 = update_H(B,
S.ravel() / 1e12,
Y.ravel() / 1e12,
method=method,
symm=symm)
np.testing.assert_allclose(B, B4, atol=0, rtol=0)
def test_modified_gram_schmidt(n, mx, my, eps1, eps2, maxiter):
rng = np.random.RandomState(2)
tol = dict(atol=1e-6, rtol=1e-6)
mgskw = dict(eps1=eps1, eps2=eps2, maxiter=maxiter)
X = get_matrix(n, mx, rng=rng)
Xout1 = modified_gram_schmidt(X, **mgskw)
_, nxout1 = Xout1.shape
np.testing.assert_allclose(Xout1.T @ Xout1, np.eye(nxout1), **tol)
np.testing.assert_allclose(np.linalg.det(X.T @ X),
np.linalg.det(X.T @ Xout1)**2, **tol)
Y = get_matrix(n, my, rng=rng)
Xout2 = modified_gram_schmidt(X, Y, **mgskw)
_, nxout2 = Xout2.shape
np.testing.assert_allclose(Xout2.T @ Xout2, np.eye(nxout2), **tol)
np.testing.assert_allclose(Xout2.T @ Y, np.zeros((nxout2, my)), **tol)
X[:, 1] = X[:, 0]
Xout3 = modified_gram_schmidt(X, **mgskw)
_, nxout3 = Xout3.shape
assert nxout3 == nxout1 - 1
np.testing.assert_allclose(Xout2.T @ Xout2, np.eye(nxout2), **tol)
def test_symmetrize(rngstate, n, m):
rng = np.random.RandomState(rngstate)
X = get_matrix(n, m, rng=rng)
minnm = min(n, m)
Y = X[:minnm, :minnm]
wrappers['symmetrize'](Y)
np.testing.assert_allclose(Y, Y.T)
def test_exact(dim, order, eta, threepoint):
rng = np.random.RandomState(1)
tol = dict(atol=1e-4, rtol=eta**2)
poly = poly_factory(dim, order, rng=rng)
x = rng.normal(size=dim)
_, g, h = poly(x)
H = NumericalHessian(lambda x: poly(x)[:2], g0=g, x0=x,
eta=eta, threepoint=threepoint)
l1, V1, AV1 = exact(h)
l2, V2, AV2 = exact(H)
np.testing.assert_allclose(l1, l2, **tol)
np.testing.assert_allclose(np.abs(V1.T @ V2), np.eye(dim), **tol)
np.testing.assert_allclose(h @ V1, AV1, **tol)
np.testing.assert_allclose(h @ V2, AV2, **tol)
P = h + get_matrix(dim, dim, rng=rng) * 1e-3
l3, V3, AV3 = exact(H, P=P)
np.testing.assert_allclose(l1, l3, **tol)
np.testing.assert_allclose(np.abs(V1.T @ V2), np.eye(dim), **tol)
def test_mgs(rngstate, n, mx, my):
rng = np.random.RandomState(rngstate)
X = get_matrix(n, mx, rng=rng)
assert wrappers['mgs'](X, None, maxiter=1) < 0
X = get_matrix(n, mx, rng=rng)
assert wrappers['mgs'](X, None, eps2=1e10) == 0
X = get_matrix(n, mx, rng=rng)
Y = get_matrix(n, my, rng=rng)
assert wrappers['mgs'](X, Y, eps2=1e10) == 0
Y = get_matrix(n, my, rng=rng)
my2 = wrappers['mgs'](Y, None)
assert my2 >= 0
np.testing.assert_allclose(Y[:, :my2].T @ Y[:, :my2],
np.eye(my2),
atol=1e-10)
X = get_matrix(n, mx, rng=rng)
mx2 = wrappers['mgs'](X, Y)
assert mx2 >= 0
np.testing.assert_allclose(X[:, :mx2].T @ X[:, :mx2],
np.eye(mx2),
atol=1e-10)
np.testing.assert_allclose(X[:, :mx2].T @ Y[:, :my2],
np.zeros((mx2, my2)),
atol=1e-10)
assert wrappers['mgs'](X, Y[:n - 1]) < 0
def test_mppi(n, m, eps):
rng = np.random.RandomState(1)
tol = dict(atol=1e-6, rtol=1e-6)
A = get_matrix(n, m, rng=rng)
U1, s1, VT1, Ainv, nsing1 = pseudo_inverse(A.copy(), eps=eps)
A_test = U1[:, :nsing1] @ np.diag(s1) @ VT1[:nsing1, :]
np.testing.assert_allclose(A_test, A, **tol)
Ainv_test = np.linalg.pinv(A)
np.testing.assert_allclose(Ainv_test, Ainv, **tol)
nsingB = nsing1 - 1
B = U1[:, :nsingB] @ np.diag(s1[:nsingB]) @ VT1[:nsingB, :]
U2, s2, VT2, Binv, nsing2 = pseudo_inverse(B.copy(), eps=eps)
def test_skew(rngstate, scale):
rng = np.random.RandomState(rngstate)
x = rng.normal(size=(3, ))
Y = get_matrix(3, 3, rng=rng)
wrappers['skew'](x, Y, scale)
np.testing.assert_allclose(scale * np.cross(np.eye(3), x), Y)